Four new experiments for the simulator (#103)

* added Experiment "Velocity and Thickness along the glacier"

* added Experiment "Mass-Balance gradient"

* added AAR experiment

* added Balance Ratio experiment

* some small changes

* some updates to the text

Co-authored-by: Fabien Maussion <fabien.maussion@uibk.ac.at>

docs/simulator.rst

@@ -54,7 +54,23 @@ and run the model again. Is the new glacier larger or smaller than before? Why?
     :class: toggle
 
     A glacier with a wider top has a larger `accumulation area <https://en.wikipedia.org/wiki/Accumulation_zone>`_.
-    It can therefore accumulate more mass (more ice) and flow further down.
+    It can therefore accumulate more mass (more ice) in the upper part. The glacier can flow further 
+    down until melt rates become large enough to compensate for this additional ice.
+
+..  admonition:: Take home messages (advanced)
+    :class: toggle
+
+    An additional (and more advanced) observation can be done by looking at the 
+    "Accumulation Area Ratio" (AAR) of the two glaciers. In the "constant width"
+    case, the glacier area is the same above and below the ELA (equilibrium AAR = 0.5,
+    only true if the mass-balance gradient is also constant). In the "wider-top"
+    case, the AAR at equilibrium is larger than 0.5: indeed, by flowing 
+    further down valley, the glacier is loosing more mass at its terminus than 
+    at its head, albeit over a different area (width). You can extend this
+    observations by experimenting with the equilibrium states of the 
+    "getting wider" and "getting narrower" glacier shapes. See also our 
+    :ref:`glacier_aar` experiments for more about the AAR.
+
 
 
 Equilibrium Line Altitude (ELA)
@@ -89,40 +105,43 @@ complex?
 
     **The lower the ELA, the larger the equilibrium glacier**. The length,
     volume or maximal thickness are not necessarily linear functions of the
-    ELA.
+    ELA: these depend on the physical relationships between ice flow and 
+    slope, as well as the feedback between glacier elevation and mass-balance.
+
+.. _glacier_slope:
 
 Glacier slope
 ~~~~~~~~~~~~~
 
 The slope of a glacier bed is one key ingredient which determines glacier flow.
-For a introduction, visit `antarcticglaciers.org (glacier-flow)`_.
+For an introduction, visit `antarcticglaciers.org (glacier-flow)`_.
 In short: glaciers flow downslope driven by the gravitational force.
-This force can be decomposed into an along-slope component and perpenticular to the slope component (see
+This force can be decomposed into an along-slope component and perpendicular to the slope component (see
 `this illustration in wikipedia <https://en.wikipedia.org/wiki/Inclined_plane#/media/File:Free_body1.3.svg>`_).
 The along-slope component "pulls" the glacier downwards and the perpendicular component "flattens" the glacier.
 
 **Experiments:**
 
-- *Beginners*: Use beginner mode with standard settings (constant width, mass
+- *Beginner*: Use beginner mode with standard settings (constant width, mass
   balance gradient of 4 and ELA of 3000) and run the model with all different
   settings for the slope and use the geometry plot for inspection.
-  Take notes on a piece of paper of the ice thickness, volume, area and length
+  Take note on a piece of paper of the ice thickness, volume, area and length
   at the end of each model run.
-- *Advanced*: Conduct the same experiment as for beginners, but additionally
+- *Advanced*: Conduct the same experiment as for Beginner, but additionally
   switch on the timeseries plot. Also take notes of the velocity and look how
   the parameters change with time in the timeseries plot.
 
 **Questions to answer:**
 
-- *Beginners*: which glaciers are thicker? Steep or flat ones? And why?
+- *Beginner*: which glaciers are thicker? Steep or flat ones? And why?
 - *Advanced*: which glaciers are faster? Steep or flat ones? How and why does
   the velocity change with time?
 
 
 ..  admonition:: Take home messages
     :class: toggle
 
-    - glaciers flow downslope
+    - glaciers flow downslope under gravity
     - the steeper the slope the thinner the glacier (larger along-slope gravitational force)
     - the flatter the slope the larger the equilibrium velocity. When the glacier
       is thin (has not much mass) the along-slope component is more important.
@@ -133,10 +152,6 @@ The along-slope component "pulls" the glacier downwards and the perpendicular co
       large and so more ice is transported downwards, and the glacier stays
       relatively thin.
 
-.. _antarcticglaciers.org (mass-balance): http://www.antarcticglaciers.org/glacier-processes/introduction-glacier-mass-balance
-.. _antarcticglaciers.org (glacier-flow): http://www.antarcticglaciers.org/glacier-processes/glacier-flow-2/glacier-flow
-
-
 Surging glaciers
 ~~~~~~~~~~~~~~~~
 
@@ -159,7 +174,7 @@ timeseries options to show the maximum velocity as well as the maximum thickness
 
 **Questions to answer:**
 
-*Beginners:*
+*Beginner:*
 
 During a surge event:
 
@@ -192,7 +207,179 @@ After a surge event:
 
 In the Notebook :ref:`notebooks_surging_glaciers` you can use OGGM to simulate surging events in Python yourself.
 
+Velocity and thickness along the glacier
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+**Experiments:**
+
+- *Beginner*: Use "Beginner mode" to simulate a glacier in equilibrium with *Width* = *Constant*, *ELA* = 3000, *Mass-balance gradient* = 4 and *Slope* = 11°.
+- *Advanced*: Use "Beginner mode" to simulate a glacier with *Width* = *Wide top, narrow bottom*, *ELA* = 3500, *Mass-balance gradient* = 4 and *Slope* = 11°.
+
+**Questions to answer:**
+
+- Make a guess as to where the ice velocity along the glacier is largest? 
+- When you made your guess, go to "Geometry opt." and tick the box *Ice velocity (top left)* and *Ice thickness (bottom left)*. 
+  Now the red/blue colors are showing the velocity/thickness distribution along the glacier. Did you guess correctly?
+- *Advanced*: what is the influence of the glacier bed bottleneck (narrowing) on ice thickness and velocity? Why?
+
+..  admonition:: Take home messages
+    :class: toggle
+
+    - *Beginner*:
+        - mass is accumulated from the top of the glacier down to the ELA (areas of positive mass-balance): all this mass must be 
+          transported downwards, and so the ice flux at equilibrium is largest at the ELA. Larger ice flux means thicker ice 
+          and faster glacier flow.
+        - below the ELA, mass is constantly ablated and the ice flux decreases: lower ice flux means thinner ice and reduced glacier flow 
+          velocity.
+    - *Advanced*:
+        - a narrowing of glacier widths means that the same amount of ice needs to be transported through a smaller door: this 
+          means that we have both the creation of a "traffic jam" (thickening) and an increase of ice velocity in order to transport 
+          more mass downwards.
+        - in this case, the maximum velocity is no longer located around the ELA but further down (at the bottleneck)
+
+Mass-balance gradient
+~~~~~~~~~~~~~~~~~~~~~
+
+See `antarcticglaciers.org (mass-balance)`_
+for an introduction about glacier mass-balance and the mass-balance gradient.
+
+In short: The climatic regime determines the glacier mass-balance gradient. Discovering global glacier 
+locations using the :ref:`explorer` reveals that glaciers can be found in quite different climates 
+around the world. Here, we will now discover how different mass-balance gradients are shaping glaciers.
+
+**Experiments:**
+
+- First, simulate a glacier in a maritime climate in temperate latitudes (larger mass-balance gradient, e.g. 10). 
+  For this, use the "Beginner mode" (*ELA* = 3000, *Width* = *Constant* and *Slope* = 11°) and let the 
+  glacier grow until it reaches equilibrium and note on a piece of paper: 
+  the equilibrium *Time*, *Length*, *Area*, *Volume*, *Max ice thickness* and *Max ice velocity* of the glacier.
+- Next, simulate a glacier in a continental climate in polar latitudes (smaller Mass Balance gradient, e.g. 3) 
+  and take some notes again.
+
+**Questions to answer:**
+
+- *Beginner*:
+    - Which of the two glaciers (maritime or continental) is thicker (*Max ice thickness*)?
+    - Which is flowing faster (*Max ice velocity*)?
+    - Which reaches the equilibrium faster (*Time*)?
+- *Advanced*:
+    - How are *Length*, *Area* and *Volume* affected?
+
+..  admonition:: Take home messages
+    :class: toggle
+
+    - the larger the mass-balance gradient, the larger the accumulation of mass (ice) at the top
+    - more accumulation leads to a thicker glacier and a larger downslope component of the gravitational force 
+      (see the :ref:`glacier_slope` experiment)
+    - this larger force causes a larger ice flux and a larger ice velocity
+    - the larger the ice velocity the faster ice is transported downwards and the faster the equilibrium is reached
+    - length and area are not much affected due to the unchanged **linear** mass-balance profile: no matter which gradient is selected, 
+      the total ice gain/loss at a certain height is only determined by the distance away from the ELA 
+      (e.g. the same amount of mass is accumulated 100 m above the ELA as there is mass ablated 100 m below the ELA, with a constant width)
+    - whereas the volume is increasing with a increasing mass-balance gradient due to a larger ice thickness
+
+.. _glacier_aar:
+
+AAR (Accumulation Area Ratio)
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+The AAR is the ratio of the accumulation area (= area above the ELA) to the total glacier area 
+(see `antarcticglaciers.org (mass-balance)`_). In this experiment we will have a look at the 
+equilibrium (or balanced) AAR (AAR-eq) and the transient (or annual) AAR (AAR-t).
+Let's make some experiments to see what the AAR can tell us about glaciers. For the interpretation 
+of the experiments, note that the total ice gain/loss at a certain elevation equals the mass-balance 
+(black line in top right figure) times the area (i.e., width) at the same elevation. This is important!
+
+**Experiments:**
+
+- *Beginner*: Use "Beginner mode" and conduct runs with *Width* = *Constant* and *Width* = *Wide top, narrow bottom*,
+  and note down the different AAR-eq (*ELA* = 3300, *Mass-balance gradient* = 4, *Slope* = 11°).
+- *Advanced*: Conduct experiments with *Constant* width and different mass-balance gradients 
+  (e.g. *Mass-balance gradient below ELA* = 4, *Mass-balance gradient above ELA* = 2 and vice versa) 
+  in "Advanced mode". Note down the different AAR-eq.
+
+**Questions to answer:**
+
+- *Beginner*:
+    - Explain the observed AAR-eq for *Constant* width and for *Wide top, narrow bottom*.
+    - For *Constant width*, what values of AAR-t (below or above 0.5) do you expect for an 
+      advancing and a retreating glacier? Can you confirm by looking at the AAR during the 
+      simulation, or using the timeseries plots. 
+- *Advanced*:
+    - How is AAR-eq changing with a different mass-balance gradients below and above the ELA?
+    - What can you conclude from the experiments about real-world glaciers which have a typical AAR-eq 
+      between 0.5 and 0.8? (see for example 
+      `Hawkins, 1985 <https://www.cambridge.org/core/journals/journal-of-glaciology/article/equilibriumline-altitudes-and-paleoenvironment-in-the-merchants-bay-area-baffin-island-nwt-canada/21991E0893BCCF88D611103F397D73D1>`_)
 
+..  admonition:: Take home messages
+    :class: toggle
+
+    - *Beginner*:
+        - In the *Constant width* case and a linear mass-balance, the AAR is around 0.5. 
+          The total ice gain/loss at a certain height is only determined by the distance away from 
+          the ELA (e.g. the same amount of mass is accumulated 100 m above the ELA as there is mass 
+          ablated 100 m below the ELA) and so the glacier area above the ELA equals the glacier area below (approximately).
+        - In the *Wide top, narrow bottom* case and a linear mass-balance, the AAR is around 0.6. 
+          In this case the total ice gain/loss at a certain height is not only determined by the distance 
+          away from the ELA but also from the width at a certain height (e.g. if the width 100 m above the ELA is double the width 
+          of 100 m below the ELA, so the total ice gain is double of the total ice loss at 100 m from the ELA). 
+          In this case the glacier length is longer compared with the case of constant width and in the lower altitudes 
+          the more negative mass-balance leads to more ice melt. Overall, the ablation area (area below ELA) stays smaller 
+          than the accumulation area, even with a longer glacier.
+        - For an advancing glacier with constant width the AAR-t is well above 0.5 (mass gain), 
+          and in the retreating case well below 0.5 (mass loss).
+    - *Advanced*:
+        - With a mass-balance gradient below the ELA twice the gradient above the ELA, the total ice loss 
+          is twice the total ice gain going the same distance away from the ELA. Therefore, the ablation area 
+          (area below the ELA) only needs to be half of the accumulation area at equilibrium. For the AAR-eq 
+          this means a value of approx. 0.6 (AAR = Ablation Area / Total Area = Ablation Area / (Accumulation Area + Ablation Area) = 
+          Ablation Area / (0.5 * Ablation Area + Ablation Area) = 1 / 1.5 = 2 / 3).
+        - For real glaciers in equilibrium with AAR between 0.5 and 0.8, we can assume wider tops and larger 
+          mass-balance gradients below the ELA.
+
+
+Balance Ratio, in the footsteps of a paleo-glaciologist
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+In this experiment we are using knowledge about Balance Ratios to estimate 
+the height of the ELA (and past climate conditions). The Balance Ratio is defined 
+as the ratio of the mass-balance gradient below the ELA to the mass-balance gradient 
+above the ELA (e.g. *Mass-balance gradient below the ELA* = 4 and *Mass-balance gradient above the ELA* = 2 gives a 
+Balance Ratio of 2). See `antarcticglaciers.org (mass-balance)`_ for an introduction about glacier
+mass-balance and the ELA, or our :ref:`glacier_basics` graphics for an illustration.
+
+In short: the height of the ELA is determined by temperature, among other things. In a warming climate, 
+the ELA is increasing.
+
+**Experiment:**
+
+    - You made expeditions to the European Alps and Kamchatka to find two glacier areas of the last glaciological maximum, 
+      by using landmarks (e.g. abrasive erosion, moraines, ...). You want to use this information to approximate the 
+      ELA height and compare the past climates at these locations (note that this experiment is only fictional).
+    - For the European Alps glacier you found an approximated past area of 3 km². The glacier geometry 
+      is a *Linear* bedrock profile with a slope of 11° and *wide top, narrow bottom* width along the glacier (typical shape for a glacier).
+    - For the Kamchatka glacier the past area was also approx. 3 km². This glacier is *getting flatter* (bedrock profile) 
+      and *getting narrower* (width along glacier).
+    - You know that a typical Balance Ratio for the European Alps is around 1.5 and for the Kamchatka around 3 
+      (e.g. `Rea, 2009 <https://www.sciencedirect.com/science/article/abs/pii/S0277379108002989?via%3Dihub>`_).
+
+**Questions:**
+    - Use the simulator and change its parameters in a "try and error" approach to find the corresponding past ELAs.
+    - Which of the two glaciers was located in a warmer environment at that time?
+    - How do different absolute values of the mass-balance gradients change your results?
+    - What additional information would be useful to know about our past glaciers in order to determine the absolute 
+      values of the mass-balance gradients?
+
+..  admonition:: Take home messages
+    :class: toggle
+
+    - Using the correct Balance Ratios, we find the following ELAs: Alps ELA = 3100 m and Kamchatka ELA = 2000
+    - From the ELA elevations, one can conclude that the past (fictional) climate in the Alps was warmer than in Kamchatka.
+    - Different magnitudes of the mass-balance gradients do not change the results a lot, but they do affect the ice thickness.
+    - Additional information about the maximum thickness could help to find the absolute gradient values.
+
+.. _antarcticglaciers.org (mass-balance): http://www.antarcticglaciers.org/glacier-processes/introduction-glacier-mass-balance
+.. _antarcticglaciers.org (glacier-flow): http://www.antarcticglaciers.org/glacier-processes/glacier-flow-2/glacier-flow
 .. _`antarcticglaciers.org (surging-glaciers)`: http://www.antarcticglaciers.org/glacier-processes/glacier-flow-2/surging-glaciers/
 .. _`this video`: http://cdn.antarcticglaciers.org/wp-content/uploads/2012/10/Panmah_and_Choktoi_glaciers_large.gif